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          LV03-10-数据结构-树
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<details class="folding-tag" blue><summary> 点击查看使用工具及版本 </summary>
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              <table>    <tr>        <td align="center" width=150px>Windows</td>        <td align="left">windows11</td>    </tr>    <tr>        <td align="center">Ubuntu</td>        <td align="left">Ubuntu16.04的64位版本</td>      </tr>    <tr>        <td align="center">VMware® Workstation 16 Pro</td>        <td align="left">16.2.3 build-19376536</td>      </tr>    <tr>        <td align="center">SecureCRT</td>        <td align="left">Version 8.7.2 (x64 build 2214)   -   正式版-2020年5月14日</td>      </tr>    <tr>        <td align="center">开发板</td>        <td align="left">正点原子 i.MX6ULL Linux阿尔法开发板</td>      </tr>    <tr>        <td align="center">uboot</td>        <td align="left">NXP官方提供的uboot，NXP提供的版本为uboot-imx-rel_imx_4.1.15_2.1.0_ga(使用的uboot版本为U-Boot 2016.03)</td>      </tr>    <tr>        <td align="center">linux内核</td>        <td align="left">linux-4.15(NXP官方提供)</td>      </tr>    <tr>        <td align="center">STM32开发板</td>        <td align="left">正点原子战舰V3(STM32F103ZET6)</td>      </tr></table>
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<h1 id="1-树的定义"><a href="#1-树的定义" class="headerlink" title="1.树的定义"></a><font size=3>1.树的定义</font></h1><p>树是一种非线性存储结构，存储的是具有一对多关系的数据元素的集合。树的一般形式如下图所示：</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502073832348.png" alt="image-20220502073832348" style="zoom:33%;" />



<p>任何一个非空树需要满足两个条件：</p>
<p>（1）<strong>有且仅有一个</strong>特定的称为根（ Root ）的节点；</p>
<p>（2）其余的节点可以分为 m （ m ≥ 0 ）个互不相交的有限集合 T1 、 T2 、……、 Tm ，其中每一个集合又是一棵树，并称为其根的子树。</p>
<h1 id="2-树的表示"><a href="#2-树的表示" class="headerlink" title="2.树的表示"></a><font size=3>2.树的表示</font></h1><p>树的表示方法有一般表示法，文氏图表示法，凹入表示法和嵌套括号法。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502080012595.png" alt="image-20220502080012595" style="zoom:33%;" />

<h1 id="3-基本概念"><a href="#3-基本概念" class="headerlink" title="3.基本概念"></a><font size=3>3.基本概念</font></h1><h2 id="3-1结点"><a href="#3-1结点" class="headerlink" title="3.1结点"></a><font size=3>3.1结点</font></h2><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502083225413.png" alt="image-20220502083225413" style="zoom:33%;" />

<ul>
<li><strong>结点</strong>：使用树结构存储的每一个数据元素都被称为结点，如图中的 A ~ M 。</li>
<li><strong>父结点</strong>：上图中，对于结点 A 、 B 、 C 、 D 来说， A 就叫做 B 、 C 、 D 的父结点，也叫双亲节点。</li>
<li><strong>子结点</strong>：上图中，对于结点 A 、 B 、 C 、 D 来说， B 、 C 、 D 就叫做 A 的子结点，也叫孩子结点。</li>
<li><strong>兄弟节点</strong>：上图中，对于结点 A 、 B 、 C 、 D 来说， B 、 C 、 D 具有相同的父结点，所以它们互为兄弟结点。</li>
<li><strong>根节点</strong>：每一个非空树都有且只有一个被称为根的结点，如上图中的 A 。<strong>判别依据</strong>：若一个结点没有父结点，那么这个结点就是整棵树的根结点。</li>
<li><strong>叶子结点</strong>：上图中，对于结点 K 、 L 、 F 、 G 、 M 、 I 、 J ，它们 没有任何子结点，那么这些结点就叫做叶子结点，也叫终端结点（叶子结点的度为 0 ）。</li>
<li><strong>分支结点</strong>：度数不为零的节点称为分支节点。</li>
<li><strong>内部结点</strong>：除根节点外的分支节点称为内部节点。</li>
</ul>
<h2 id="3-2空树与子树"><a href="#3-2空树与子树" class="headerlink" title="3.2空树与子树"></a><font size=3>3.2空树与子树</font></h2><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502083528569.png" alt="image-20220502083528569" style="zoom:33%;" />

<ul>
<li><strong>空树</strong>：若集合为空，那么它将可以构成一个没有结点的空树。</li>
<li><strong>子树</strong>：上图中，整棵树的根节点为 A ，只看 B 、 E 、 F 、 K 、 L ，那么这也是一个树的结构，并且结点 B 为根结点，所以 B 、 E 、 F 、 K 、 L 组成的树称为整个树的子树。同样， E 、 K 、 L 也构成一颗子树，根结点为 E ； C 、 G 也构成子树，根节点为 C 。</li>
</ul>
<p>【注意】</p>
<p>（1）单个结点也是一棵树，且根节点为结点本身。</p>
<p>（2）在树的结构中，对于同一个结点的各个子树之间不可以有交集，若有，那将不符合树的结构了。</p>
<h2 id="3-3度和层次"><a href="#3-3度和层次" class="headerlink" title="3.3度和层次"></a><font size=3>3.3度和层次</font></h2><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502083759939.png" alt="image-20220502083759939" style="zoom:33%;" />

<ul>
<li><p><strong>结点的度</strong>：对于树的一个结点所拥有子树的个数称之为结点的<strong>度</strong>( Degree )，度也可以说是一个结点下有多少个分支。例如上图中的结点 A ，它下边有三个子树（三个分支），所以结点 A 的度为 3 ；对于结点 M ，度为 0 。</p>
</li>
<li><p><strong>树的度</strong>：一棵树的度数是指这个树中结点的<strong>最大</strong>度数，上图中，各个结点度的最大值为 3 ，所以这颗树的度数为 3 。</p>
</li>
<li><p><strong>结点层次</strong>：从树根开始，树根所在层为第 1 层，根的孩子结点所在的层为第 2 层，如图中所示。 A  结点在第 1 层， B 、 C 、 D  为第 2 层， E 、 F 、 G 、 H 、 I 、 J  在第 3 层， K 、 L 、 M  在第 4 层。</p>
</li>
<li><p><strong>堂兄弟</strong>：双亲在同一层的结点互为堂兄，图中结点 G 与 E 、 F 、 H 、 I 、 J 互为堂兄弟。</p>
</li>
<li><p><strong>树的深度</strong>：一棵树的深度（高度）是树中结点所在的最大的层次，上图中树的高度为 4 。</p>
</li>
</ul>
<p>【注意】</p>
<p>（1）结点的层次等于这个结点的父结点层数加 1 。</p>
<p>（2）根节点层数定义为 1 。</p>
<h2 id="3-4路径与祖先"><a href="#3-4路径与祖先" class="headerlink" title="3.4路径与祖先"></a><font size=3>3.4路径与祖先</font></h2><img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502085149575.png" alt="image-20220502085149575" style="zoom:33%;" />

<ul>
<li><strong>路径</strong>：一个结点系列$k_1, k_2, …… , k_i,k_{i+1},  ……, k_j$并满足$k_i$是$k_{i+1}$的父节点，就称为一条从$k_1$到$k_j$的路径。路径的<strong>长度</strong>为 j-1 ,即路径中的<strong>边数</strong>。</li>
<li><strong>祖先与子孙</strong>：路径中前面的节点是后面节点的祖先，后面节点是前面节点的子孙。</li>
</ul>
<h2 id="3-5有序树和无序树"><a href="#3-5有序树和无序树" class="headerlink" title="3.5有序树和无序树"></a><font size=3>3.5有序树和无序树</font></h2><p>若树中每个节点的各个子树的排列为从左到右，不能交换，即兄弟之间是有序的，则该树称为<strong>有序树</strong>。反之称为<strong>无序树</strong>。</p>
<img data-src="https://fanhua-picture.oss-cn-hangzhou.aliyuncs.com/01%E5%B5%8C%E5%85%A5%E5%BC%8F%E5%BC%80%E5%8F%91/01HQ%E8%AF%BE%E7%A8%8B%E4%BD%93%E7%B3%BB/LV03-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84/LV03-10-%E6%95%B0%E6%8D%AE%E7%BB%93%E6%9E%84-%E6%A0%91/img/image-20220502085927073.png" alt="image-20220502085927073" style="zoom:33%;" />

<p>如上图，如果是其本身是一棵有序树，则以结点  B  为根结点的子树为整棵树的第一个孩子，以结点  D  为根结点的子树为整棵树的最后一个孩子。</p>
<h2 id="3-6森林"><a href="#3-6森林" class="headerlink" title="3.6森林"></a><font size=3>3.6森林</font></h2><p>由  m （ m &gt;&#x3D; 0 ）个互不相交的树组成的集合被称为<strong>森林</strong>。树可以理解为是由根结点和若干子树构成的，而这若干子树本身是一个森林，所以，树还可以理解为是由根结点和森林组成的。用一个式子表示为：</p>
<figure class="highlight c"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">Tree =（root,F）</span><br><span class="line">    </span><br><span class="line">root 表示树的根结点，</span><br><span class="line">F 表示由 m（m &gt;= <span class="number">0</span>）棵树组成的森林。</span><br></pre></td></tr></table></figure>

<p>【注意】树去掉根节点就成为森林，森林加上一个新的根节点就成为树。</p>
<h2 id="3-7树的逻辑结构"><a href="#3-7树的逻辑结构" class="headerlink" title="3.7树的逻辑结构"></a><font size=3>3.7树的逻辑结构</font></h2><p>树中任何节点都可以有零个或多个直接后继节点（子节点），但至多只有一个直接前趋节点（父节点）。</p>
<p>根节点没有前趋节点，叶节点没有后继节点。 </p>
<h1 id="4-基本性质"><a href="#4-基本性质" class="headerlink" title="4.基本性质"></a><font size=3>4.基本性质</font></h1><p>（1）树的结点数等于所有结点度数加 1 。</p>
<blockquote>
<p>【证明】每个结点的度数分别对应这个结点的子结点，假设所有结点的度数为b，那么结点数n &#x3D; b + root &#x3D; b +1。</p>
</blockquote>
<p>（2）度为 m 的树中，第 i 层上至多有$m^{i-1}$个结点（ i ≥ 1 ）。</p>
<blockquote>
<p>【证明】由数学归纳法证明</p>
<p>1.当树为第1层时，第1层至多有$m^0&#x3D;1$个结点;<br>2.当树为第 i-1 层时，由该性质得第 i - 1层至多有$m^{i-2}$个结点;<br>3.当树为第 i 层时，由于树的度为 m ，可知每个第 i - 1 层的每个结点至多有 m 个子结点，因此第 i 层至多有$m^{i-2}*m&#x3D;m^{i-1}$个结点。<br>由数学归纳法可知，上述结论正确。</p>
</blockquote>
<p>（3）高度为 h 的 m 叉树至多有$(m^{h-1})&#x2F;(m-1)$个结点。</p>
<blockquote>
<p>【证明】</p>
<p>由性质(1)可知，度为 m 的树第 i 层至多有$m{i-1}$个结点；</p>
<p>因此高度为 h 的 m 叉树至多有 $n &#x3D; m^0+m^1+…+m^{h-1}$个结点；</p>
<p>用等比数列求和公式，公比为 m ，可以得到结点数$n&#x3D; (m^h-1)&#x2F;(m-1)$。</p>
</blockquote>
<p>（3）具有 n 个结点的 m 叉树的最小高度为$[\log_m[n*(m-1)+1]]$</p>
<blockquote>
<p>【证明】</p>
<p>由性质(2)逆推，要得到 m 叉树的最小高度，那么每一层都具有最多的结点，由于高度为整数，且多余结点也是一层，所以树的高度需要向上取整。</p>
</blockquote>

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